cramers rule

Dear romil gayo,

Cramer’s rule is a method for solving linear simultaneous equations. It makes use of determinants. 

In linear algebra, Cramer''s rule is a theorem, which gives an expression for the solution of a system of linear equations with as many equations as unknowns, valid in those cases where there is a unique solution.

Eg : 

To find rules (or formulas) that may be used solve any system of equations, we need to solve the general system of the form 

a ₁ x + b ₁ y = c ₁    (1)
a ₂ x + b ₂ y = c ₂    (2)

We multiply equation (1) by b ₂ and equation (2) by -b ₁ and add the right and left hand terms. 

b ₂ a ₁ x + b ₂ b ₁ y = b ₂ c ₁    (1)
- b ₁ a ₂ x - b ₁ b ₂ y = - b ₁ c ₂    (2)____________________________
b ₂ a ₁ x - b ₁ a ₂ x = b ₂ c ₁ - b ₁ c ₂
Assuming that a ₁ b ₂ - a ₂ b ₁ is not equal to zero, solve the above equation for x to obtain. 

x = ( c ₁ b ₂ - c ₂ b ₁ ) / ( a ₁ b ₂ - a ₂ b ₁ )
We can use similar steps to eliminate x and solve for y to obtain. 

y = ( a ₁ c ₂ - a ₂ c ₁ ) / ( a ₁ b ₂ - a ₂ b ₁ )

Using the determinant of a Matrix notation, the solution to the given 2 by 2 system of linear equations is given by 

x = D _x / D and y = D _y / D 

where D, D _x and D _y are the determinants defined by 



For a 3 by 3 system of linear equations of the form 

a ₁ x + b ₁ y + c ₁ z = d ₁    (1)
a ₂ x + b ₂ y + c ₂ z = d ₂    (2)
a ₃ x + b ₃ y + c ₃ z = d ₃    (3)

Cramer''s rule gives the solution as follows 

x = D _x / D , y = D _y / D and z = D _z / D 

where D, D _x, D _y and D _z are determinants defined by 

Hope this helped you immensely..!

All the Very Best & Good Luck to you ..!!

Best Regards,

AskIITians Expert,

Godfrey Classic Prince

IIT-M

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